Today's computational and experimental paradigms feature complex models along with disparate and, frequently, enormous data sets. This necessitates the development of theoretical and computational strategies for efficient and robust numerical algorithms that effectively resolve the important features and characteristics of these complex computational models. The desiderata for resolving the underlying model features is often application-specific and combines mathematical tasks like approximation, prediction, calibration, design, and optimization. Running simulations that fully account for the variability of the complexities of modern scientific models can be infeasible due to the curse of dimensionality, chaotic behavior or dynamics, and/or overwhelming streams of informative data.
This semester program focuses on both theoretical investigation and practical algorithm development for reduction in the complexity - the dimension, the degrees of freedom, the data - arising in these models. The four broad thrusts of the program are (1) Mathematics of reduced order models, (2) Algorithms for approximation and complexity reduction, (3) Computational statistics and data-driven techniques, and (4) Application-specific design. The particular topics include classical strategies such as parametric sensitivity analysis and best approximations, mature but active topics like principal component analysis and information-based complexity, and promising nascent topics such as layered neural networks and high-dimensional statistics.
This program will integrate diverse fields of mathematical analysis, statistical sciences, data and computer science, and specifically attract researchers working on model order reduction, data-driven model calibration and simplification, computations and approximations in high dimensions, and data-intensive uncertainty quantification. Various workshops will be designed to stimulate interaction between these research areas and establish cross-disciplinary collaboration. Investigation and assimilation of complementary approaches through other program events will achieve cross-fertilization and serve as a nexus for multiple research communities.
Justin Baker, Elena Cherkaev, Akil Narayan, Bao Wang, Learning POD of Complex Dynamics Using Heavy-ball Neural ODEs, baker2023learning:2202.12373, 2023.
Petar Mlinarić, Serkan Gugercin, A Unifying Framework for Interpolatory \(\boldsymbol\mathcalL_2
\)-Optimal Reduced-Order Modeling, SIAM Journal on Numerical Analysis 61 (2023) no. 5, 2133–2156.
Petar Mlinarić, Serkan Gugercin, $\mathcalL_2$-optimal Reduced-order Modeling Using Parameter-separable Forms, mlinaric2022mathcall2optimal:2206.02929, 2022.
Peter Benner, Serkan Gugercin, Steffen W. R. Werner, A unifying framework for tangential interpolation of structured bilinear control systems, Numerische Mathematik 155 (2023) no. 3–4, 445–483.
Misha E. Kilmer, Arvind K. Saibaba, Structured Matrix Approximations via Tensor Decompositions, kilmer2021structured:2105.01170, 2021.
Yiming Xu, Akil Narayan, Hoang Tran, Clayton G. Webster, Analysis of The Ratio of $\ell_1$ and $\ell_2$ Norms in Compressed Sensing, xu2021analysis:2004.05873, 2021.
Jarom D. Hogue, Robert M. Kirby, Akil Narayan, Weight Matrix Dimensionality Reduction in Deep Learning via Kronecker Multi-layer Architectures, hogue2023weight:2204.04273, 2023.
Michael Penwarden, Shandian Zhe, Akil Narayan, Robert M. Kirby, Multifidelity modeling for Physics-Informed Neural Networks (PINNs), Journal of Computational Physics 451 (2022), 110844.
Vahid Keshavarzzadeh, Robert M. Kirby, Akil Narayan, Variational inference for nonlinear inverse problems via neural net kernels: Comparison to Bayesian neural networks, application to topology optimization, Computer Methods in Applied Mechanics and Engineering 400 (2022), 115495.
Nuojin Cheng, Osman Asif Malik, Yiming Xu, Stephen Becker, Alireza Doostan, Akil Narayan, Quadrature Sampling of Parametric Models with Bi-fidelity Boosting, cheng2022quadrature:2209.05705, 2022.
Ion Victor Gosea, Serkan Gugercin, Steffen W. R. Werner, Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems, gosea2023structured:2303.12576, 2023.
Boris Kramer, Serkan Gugercin, Jeff Borggaard, Nonlinear Balanced Truncation: Part 2 -- Model Reduction on Manifolds, kramer2023nonlinear:2302.02036, 2023.
Boris Kramer, Serkan Gugercin, Jeff Borggaard, Linus Balicki, Nonlinear Balanced Truncation: Part 1-Computing Energy Functions, kramer2022nonlinear:2209.07645, 2022.
Zhichao Peng, Yanlai Chen, Yingda Cheng, Fengyan Li, A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation, peng2022micromacro:2211.04677, 2022.
Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan, Model reduction for fractional elliptic problems using Kato's formula, dinh2019model:1904.09332, 2019.
Yanlai Chen, Sigal Gottlieb, Lijie Ji, Yvon Maday, An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation, chen2021eimdegradation:2101.05902, 2021.
Nuojin Cheng, Osman Asif Malik, Yiming Xu, Stephen Becker, Alireza Doostan, Akil Narayan, Quadrature Sampling of Parametric Models with Bi-fidelity Boosting, arXiv preprint arXiv:2209.05705 (2022).
Daniele Venturi, Alec Dektor, Spectral methods for nonlinear functionals and functional differential equations, Research in the Mathematical Sciences 8 (2021) no. 2, 27.
Jarom D Hogue, Robert M Kirby, Akil Narayan, Dimensionality Reduction in Deep Learning via Kronecker Multi-layer Architectures, arXiv preprint arXiv:2204.04273 (2022).
Michael Penwarden, Shandian Zhe, Akil Narayan, Robert M Kirby, Multifidelity modeling for physics-informed neural networks (PINNs), Journal of Computational Physics 451 (2022), 110844.
Justin Baker, Elena Cherkaev, Akil Narayan, Bao Wang, Learning Proper Orthogonal Decomposition of Complex Dynamics Using Heavy-ball Neural ODEs, Journal of Scientific Computing 95 (2023) no. 2, 54.
Vahid Keshavarzzadeh, Robert M Kirby, Akil Narayan, Variational inference for nonlinear inverse problems via neural net kernels: Comparison to Bayesian neural networks, application to topology optimization, Computer Methods in Applied Mechanics and Engineering 400 (2022), 115495.
Misha E Kilmer, Arvind K Saibaba, Structured Matrix Approximations via Tensor Decompositions, arXiv preprint arXiv:2105.01170 (2021).
Akil Narayan, Liang Yan, Tao Zhou, Optimal design for kernel interpolation: Applications to uncertainty quantification, Journal of Computational Physics 430 (2021), 110094.
Yanlai Chen, Lijie Ji, Akil Narayan, Zhenli Xu, L1-based reduced over collocation and hyper reduction for steady state and time-dependent nonlinear equations, Journal of Scientific Computing 87 (2021) no. 1, 1-21.
Elizabeth Qian, Jemima M Tabeart, Christopher Beattie, Serkan Gugercin, Jiahua Jiang, Peter R Kramer, Akil Narayan, Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference, Journal of Scientific Computing 91 (2022) no. 1, 1-30.
Peter Benner, Serkan Gugercin, Steffen WR Werner, Structure-preserving interpolation for model reduction of parametric bilinear systems, Automatica 132 (2021), 109799.
Ion Victor Gosea, Serkan Gugercin, The AAA framework for modeling linear dynamical systems with quadratic output, arXiv preprint arXiv:2005.10316 (2020).
Andrea Carracedo Rodriguez, Serkan Gugercin, The p-AAA algorithm for data driven modeling of parametric dynamical systems, arXiv preprint arXiv:2003.06536 (2020).
Vahid Keshavarzzadeh, Robert M Kirby, Akil Narayan, Multilevel designed quadrature for partial differential equations with random inputs, SIAM Journal on Scientific Computing 43 (2021) no. 2, A1412-A1440.
Dihan Dai, Yekaterina Epshteyn, Akil Narayan, Non-Dissipative and Structure-Preserving Emulators via Spherical Optimization, arXiv preprint arXiv:2108.12053 (2021).
Vahid Keshavarzzadeh, Robert M Kirby, Akil Narayan, Robust topology optimization with low rank approximation using artificial neural networks, Computational Mechanics 68 (2021) no. 6, 1297-1323.
Roland Pulch, Akil Narayan, Tatjana Stykel, Sensitivity analysis of random linear differential--algebraic equations using system norms, Journal of Computational and Applied Mathematics 397 (2021), 113666.
Ling Guo, Akil Narayan, Yongle Liu, Tao Zhou, Sparse approximation of data-driven Polynomial Chaos expansions: an induced sampling approach, arXiv preprint arXiv:2008.10121 (2020).
Mani Razi, Robert Mike Kirby, Akil Narayan, Kernel optimization for low-rank multifidelity algorithms, International Journal for Uncertainty Quantification 11 (2021) no. 1.
Peter Benner, Serkan Gugercin, Steffen WR Werner, Structure-preserving interpolation of bilinear control systems, Advances in Computational Mathematics 47 (2021) no. 3, 1-38.
Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan, Model reduction for fractional elliptic problems using Kato's formula, arXiv preprint arXiv:1904.09332 (2019).
Brian B Avants, Nicholas J Tustison, James R Stone, Interpretable, similarity-driven multi-view embeddings from high-dimensional biomedical data, arXiv preprint arXiv:2006.06545 (2020).
Gang Chen, Peter B Monk, Yangwen Zhang, L\^∞ Norm Error Estimates for HDG Methods Applied to the Poisson Equation with an Application to the Dirichlet Boundary Control Problem, SIAM Journal on Numerical Analysis 59 (2021) no. 2, 720-745.
Vladimir Druskin, Shari Moskow, Mikhail Zaslavsky, Lippmann--Schwinger--Lanczos algorithm for inverse scattering problems, Inverse Problems 37 (2021) no. 7, 075003.
Kelly M Diamond, Brian B Avants, A Murat Maga, Machine learning-based segmentation and landmarking of 2D fish images, INTEGRATIVE AND COMPARATIVE BIOLOGY, vol. 61, OXFORD UNIV PRESS INC JOURNALS DEPT, 2001 EVANS RD, CARY, NC 27513 USA, 2021, pp. E1100-E1101.
Davide Palitta, Matrix equation techniques for certain evolutionary partial differential equations, Journal of Scientific Computing 87 (2021) no. 3, 1-36.
Sergiy Borodachov, Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere, arXiv preprint arXiv:2203.13756 (2022).
Christian Himpe, Sara Grundel, Peter Benner, Model order reduction for gas and energy networks, Journal of Mathematics in Industry 11 (2021) no. 1, 1-46.
Vladimir Druskin, Stefan G{\"u}ttel, Leonid Knizhnerman, Model order reduction of layered waveguides via rational Krylov fitting, BIT Numerical Mathematics (2022), 1-22.
Lihong Feng, Peter Benner, On error estimation for reduced-order modeling of linear non-parametric and parametric systems, ESAIM: Mathematical Modelling and Numerical Analysis 55 (2021) no. 2, 561-594.
Vladimir Druskin, Shari Moskow, Mikhail Zaslavsky, On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays, arXiv preprint arXiv:2112.09634 (2021).
Peter Benner, Pawan Goyal, Jan Heiland, Igor Pontes Duff, Operator inference and physics-informed learning of low-dimensional models for incompressible flows, arXiv preprint arXiv:2010.06701 (2020).
Nihar Sawant, Boris Kramer, Benjamin Peherstorfer, Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference, arXiv preprint arXiv:2107.02597 (2021).
Jacob Y Hesterman, Elliot Greenblatt, Andrew Novicki, Ali Ghayoor, Tyler Wellman, Brian Avants, Practical applications of machine learning in imaging trials, Visualizing and Quantifying Drug Distribution in Tissue V, vol. 11624, International Society for Optics and Photonics, 2021, pp. 116240I.
Sofia Davydycheva, Vladimir Druskin, Leonid Knizhnerman, Michael Rabinovich, Quality control of ultra-deep resistivity imaging using fast 3D electromagnetic modeling, SEG Technical Program Expanded Abstracts 2020, Society of Exploration Geophysicists, 2020, pp. 380-384.
Petar Mlinari{\'c}, Stephan Rave, Jens Saak, Parametric model order reduction using pyMOR, Model Reduction of Complex Dynamical Systems, Springer, 2021, pp. 357-367.
Julian Henning, Davide Palitta, Valeria Simoncini, Karsten Urban, Very Weak Space-Time Variational Formulation for the Wave Equation: Analysis and Efficient Numerical Solution, arXiv preprint arXiv:2107.12119 (2021).
Brian B Avants, Nicholas J Tustison, James R Stone, Similarity-driven multi-view embeddings from high-dimensional biomedical data, Nature computational science 1 (2021) no. 2, 143-152.
Kui Ren, Yimin Zhong, Unique determination of absorption coefficients in a semilinear transport equation, SIAM Journal on Mathematical Analysis 53 (2021) no. 5, 5158-5184.
Nicholas J Tustison, Philip A Cook, Andrew J Holbrook, Hans J Johnson, John Muschelli, Gabriel A Devenyi, Jeffrey T Duda, Sandhitsu R Das, Nicholas C Cullen, Daniel L Gillen, others, The ANTsX ecosystem for quantitative biological and medical imaging, Scientific reports 11 (2021) no. 1, 1-13.
Fatoumata Sanogo, Carmeliza Navasca, Stefan Kindermann, Tensor Deblurring and Denoising Using Total Variation, arXiv preprint arXiv:2111.03965 (2021).
Brian Avants, Elliot Greenblatt, Jacob Hesterman, Nicholas Tustison, Deep Volumetric Feature Encoding for Biomedical Images, International Workshop on Biomedical Image Registration, Springer, 2020, pp. 91-100.
Zhuo-Heng He, Carmeliza Navasca, Xiang-Xiang Wang, Decomposition for a Quaternion Tensor Triplet with Applications, Advances in Applied Clifford Algebras 32 (2022) no. 1, 1-19.
Zhichao Peng, Min Wang, Fengyan Li, A learning-based projection method for model order reduction of transport problems, arXiv preprint arXiv:2105.14633 (2021).
Peter Benner, Heike Fassbender, Philip Saltenberger, A rational Even-IRA algorithm for the solution of T-even polynomial eigenvalue problems, SIAM Journal on Matrix Analysis and Applications 42 (2021) no. 3, 1172-1198.
Liliana Borcea, Vladimir Druskin, J{\"o}rn Zimmerling, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientific Computing 89 (2021) no. 1, 1-36.
Sridhar Chellappa, Lihong Feng, Peter Benner, A training set subsampling strategy for the reduced basis method, Journal of Scientific Computing 89 (2021) no. 3, 1-34.
Neeraj Sarna, Jan Giesselmann, Peter Benner, Data-driven snapshot calibration via monotonic feature matching, arXiv preprint arXiv:2009.08414 (2020).
Neeraj Sarna, Peter Benner, Data-Driven model order reduction for problems with parameter-dependent jump-discontinuities, Computer Methods in Applied Mechanics and Engineering 387 (2021), 114168.
Peter Benner, Sara Grundel, Petar Mlinari{\'c}, Clustering-Based Model Order Reduction for Nonlinear Network Systems, Model Reduction of Complex Dynamical Systems, Springer, 2021, pp. 75-96.
Davide Palitta, Sanda Lefteriu, An efficient, memory-saving approach for the Loewner framework, Journal of Scientific Computing 91 (2022) no. 2, 1-25.
Wayne Isaac Tan Uy, Yuepeng Wang, Yuxiao Wen, Benjamin Peherstorfer, Active operator inference for learning low-dimensional dynamical-system models from noisy data, arXiv preprint arXiv:2107.09256 (2021).
Vladimir Druskin, Alexander V Mamonov, Mikhail Zaslavsky, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Journal of Scientific Computing 90 (2022) no. 1, 1-30.
James R Stone, Brian B Avants, Nicholas J Tustison, Eric M Wassermann, Jessica Gill, Elena Polejaeva, Kristine C Dell, Walter Carr, Angela M Yarnell, Matthew L LoPresti, others, Functional and structural neuroimaging correlates of repetitive low-level blast exposure in career breachers, Journal of neurotrauma 37 (2020) no. 23, 2468-2481.
Dihan Dai, Yekaterina Epshteyn, Akil Narayan, Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations, Journal of Computational Physics 452 (2022), 110901.
Yiming Xu, Akil Narayan, Budget-limited distribution learning in multifidelity problems, arXiv preprint arXiv:2105.04599 (2021).
Yiming Xu, Vahid Keshavarzzadeh, Robert M Kirby, Akil Narayan, A bandit-learning approach to multifidelity approximation, SIAM Journal on Scientific Computing 44 (2022) no. 1, A150-A175.
Ryleigh A Moore, Akil Narayan, Adaptive Density Tracking by Quadrature for Stochastic Differential Equations, arXiv preprint arXiv:2105.08148 (2021).
Yiming Xu, Akil Narayan, Hoang Tran, Clayton G Webster, Analysis of the ratio of ℓ1 and ℓ2 norms in compressed sensing, Applied and Computational Harmonic Analysis 55 (2021), 486-511.
Ion Victor Gosea, Serkan Gugercin, Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework, Journal of Scientific Computing 91 (2022) no. 1, 1-28.
Ion Victor Gosea, Serkan Gugercin, Christopher Beattie, Data-driven balancing of linear dynamical systems, SIAM Journal on Scientific Computing 44 (2022) no. 1, A554-A582.
Lihong Feng, Guosheng Fu, Zhu Wang, A FOM/ROM Hybrid Approach for Accelerating Numerical Simulations, Journal of Scientific Computing 89 (2021) no. 3, 1-16.
Zhichao Peng, Yanlai Chen, Yingda Cheng, Fengyan Li, A reduced basis method for radiative transfer equation, Journal of Scientific Computing 91 (2022) no. 1, 1-27.
